A dual band antenna disclosed in U.S. patent application Ser. No. 09/206,445, has a coil antenna element with a first winding at a feed point, and a second winding at a far end of the antenna. A reactive or parasitic antenna element is provided on a film that forms a wrapping over the coil. The film provides a thin dielectric between the coil and the reactive element, which capacitively couples the coil and the reactive element. At lower frequencies, the reactive element is electrically inactive, while at higher frequencies, the element establishes a short circuit.